We resolve three interrelated problems on reduced Kronecker coefficients g ¯(α,β,γ). First, we disprove the saturation property which states that g ¯(Nα,Nβ,Nγ)>0 implies g ¯(α,β,γ)>0 for all N>1. Second, we esimate the maximal g ¯(α,β,γ), over all |α|+|β|+|γ|=n. Finally, we show that computing g ¯(λ,μ,ν) is strongly #P-hard, i.e. #P-hard when the input (λ,μ,ν) is in unary.